Infinitesimal calculus

The infinitesimal calculus (or differential and integral calculus ) is a branch of the Mathématiques, developed starting from the Algèbre and of the Géométrie, which implies two complementary major ideas:

• the differential calculus is the theory which treats rates of Variation and utilizes the method of differentiation. In term of mathematical functions, the Speed, the Acceleration, and the slopes of the Courbe S in a given point can all be described on common a symbolic system basis.
• the Integral calculus , which develops the idea of integration, utilizes the concept of surface underlain by the Graphe of a function and includes related concepts like the Volume.

These two concepts define operations opposite in the precise direction defined by the fundamental Théorème S of the infinitesimal calculus. This wants to say that they have an equivalent priority. However the usual teaching approach starts with differential calculus.

History

See also: History of the infinitesimal calculus

The development of the infinitesimal calculus is allotted to Archimedes, Leibniz and Newton. However, when the infinitesimal calculus was initially developed, a controversy was raised on which had paternity of it; Leibniz and Newton being principal candidates. The truth will probably never be known and in any event it imports little nowadays. The major contribution of Leibniz was without question its marking system.

The controversy was however unhappy because it divided during many years the mathematicians Anglophone S and those of the remainder of the Europe. That delayed the progress of the analyzes (mathematical based on the infinitesimal calculus) in Great Britain for a long time. The terminology and the notations of Newton were clearly less flexible than those of Leibniz. They in spite of were very preserved until the beginning of the XIXe century when the work of the Analytical Society successfully introduced the notation of Leibniz in Great Britain.

It is thought that Newton discovered several concepts quite front Leibniz, but that this last was the first to publish them. Currently, it is considered that Leibniz and Newton developed the infinitesimal calculus independently.

Barrow, Descartes, Fermat, Huygens and Wallis contributed also to a lesser extent to the development of the infinitesimal calculus.

Kowa Seki, a mathematician Japanese contemporary of Leibniz and Newton, also stated some basic principles of the integral calculus. However its work was not known in Occident at that time following the lack of contacts with the the Far East.

The justification first of the development of differential calculus was to find a solution of the “problem of the tangent ”.

Differential calculus

See also: Derived

Differential calculus consists in finding the instantaneous Rates of variation (or derived ) of the value of a function compared to the variations of (of) the parameter (S) of this one. This concept is in the middle of many problems of Physique. For example, the basic theory of the electrical circuits is formulated in term of differential equations to describe the oscillating systems.

The derivative of a function makes it possible to find its extrema (minimum and maximum) by studying its variations. Another application of differential calculus is the Méthode of Newton, a algorithm which makes it possible to find the Zero S of a function by locally approaching it by its tangent S. This is only one very short outline of the many applications of the infinitesimal calculus in problems which at first sight are not formulated in these terms.

Some allot to Fermat the paternity of differential calculus.

Integral calculus

See also: Integral

The integral calculus studies the methods making it possible to find the Intégrale of a function. It can be defined as the limit sum of terms which correspond each one to surface of a fine strip underlain by the graph of the function. Thus defined, integration gives an average manpower to calculate the surface under a curve as well as the surface and the volume of solids like the Sphère or the cone.

Bases

The conceptual bases of the infinitesimal calculus include the concepts of limiting functions, , continuation S infinite, series S infinite and Continuité. Its tools include the techniques of handling symbolic system associated with the Elementary algebra and the mathematical induction. The modern version of the infinitesimal calculus is known as real Analyze which consists of a rigorous derivation of the results of the infinitesimal calculus like in generalizations like the theory of measurement and the analyzes functional.

Fundamental theorem of the analysis

See also: fundamental Theorem of the analysis

The fundamental Théorème of the analysis watch that the differentiation and integration are, in a certain direction, operations opposite. It is this “discovery” by Newton and Leibniz which is at the origin of the explosion of the analytical results. This bond enables us to find the total variation of a function on an interval from its instantaneous variation, by integrating the latter. The fundamental theorem gives us also a method to calculate much definite integrals in an algebraic way, without really passing in extreme cases, by finding the Primitive. It also enables us to solve certain differential equations. A differential equation is an equation which binds a function has its derivative. The differential equations are fundamental in science.

Applications

To make concrete these concepts, let us consider in the plan (xOy) a rectangle on side X and Y. Its surface is equal to xy and depends on coordinates X and there on the point Mr. While following an intuitive step, one agrees to note by dx a very small variation of the variable X . When one subjects the point M a very weak displacement, surface will change and one can write that S+dS= (x+dx). (y+dy) =x.y +x.dy+y.dx + dx.dy, and one from of deduced easily that dS= y.dx+x.dy+dx.dy.

A simple numerical application where X and would be meters there and dx and famous Dy of the centimetres that dx.dy is negligible compared to the other sizes

One can give a precise mathematical statute to the notations dx and Dy (which is differential forms), and to the quantity dx.dy which is then of the second order . Preceding calculation is in fact a calculation of Développement limited to order 1, utilizing the derivative first function xy compared to the two variables.

One thus writes:

$dS=\; (x+dx).\; (y+dy)\; -\; x.y\; =y.dx\; +\; x.dy\; =\; (there,\; X)\; \backslash \; cdot\; (dx,\; Dy)\; =\; \backslash \; overrightarrow\; \backslash \; nabla\; S\; \backslash \; cdot\; \backslash \; overrightarrow\; \{dOM\}$

$\backslash \; overrightarrow\; \backslash \; nabla\; S\; \backslash \; cdot\; \backslash \; overrightarrow\; \{dOM\}\; =\; (there\; \backslash \; vec\; I\; +x\; \backslash \; vec\; J)\; \backslash \; cdot\; (dx\; \backslash \; vec\; i+\; Dy\; \backslash \; vec\; J)\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; partial\; (xy)\}\{\backslash \; partial\; X\}\; \backslash \; vec\; I\; +\; \backslash \; frac\; \{\backslash \; partial\; (xy)\}\{\backslash \; partial\; there\}\; \backslash \; vec\; J\; \backslash \; right)\; \backslash \; cdot\; (dx\; \backslash \; vec\; i+\; Dy\; \backslash \; vec\; J)$

All these equalities are various ways of writing… a scalar Produit of two vectors:

$dS=\; (x+dx).\; (y+dy)\; -\; x.y\; =y.dx\; +\; x.dy\; =\; \backslash \; mathrm\; \{\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\}\; (xy)\; \backslash \; cdot\; \backslash \; overrightarrow\; \{dOM\}\; =\; \backslash \; overrightarrow\; \backslash \; nabla\; (xy)\; \backslash \; cdot\; \backslash \; overrightarrow\; \{dOM\}$ where $\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; (xy)\; =\; (there,\; X)$

The interest of the introduction of these vectors to express the variation of a function of several parameters is to visualize the fact that the function will vary more in the direction of the vector Gradient and that it will not vary for any change of the parameters in a direction perpendicular to the gradient.

$(there\; \backslash \; vec\; I\; +x\; \backslash \; vec\; J)\; \backslash \; cdot\; (dx\; \backslash \; vec\; i+\; Dy\; \backslash \; vec\; J)\; =0$ for: $there\; dx\; +\; X\; Dy\; =\; 0$ in our example of the rectangle.

The development and the use of the infinitesimal calculus had important consequences in practically all the fields. It is at the base of many sciences, in particular the Physique. Almost all the techniques and modern technologies make a fundamental use of the infinitesimal calculus.

This one extended with the differential equations, the vector Calculus, the Calcul of the variations, the Analyze complexes, or the differential Géométrie.

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